Geometric function theory: Difference between revisions

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'''Geometric function theory''' is the study of [[Geometry|geometric]] properties of [[analytic function]]s. A fundamental result in the theory is the [[Riemann mapping theorem]].

==Riemann Mapping Theorem==

Let ''z''{{su|b=0}} be a point in a simply-connected region ''D''{{su|b=1}} (''D''{{su|b=1}}≠ ℂ) and ''D''{{su|b=1}} having at least two boundary points. Then there exists a unique analytic function ''w = f(z)'' mapping ''D''{{su|b=1}} bijectively into the open unit disk |''w''|<1 such that ''f(''z''{{su|b=0}})''=0 and

''Re f ′(''z''{{su|b=0}})''=0.

It should be noted that while Riemann's mapping theorem demonstrates the existence of a mapping function, it does not actually ''exhibits'' this function.

==References==

*{{cite book |title=Geometric Function Theory: Explorations in Complex Analysis|

first=Steven|last=Krantz|publisher=Springer|year=2006|isbn=0817643397}}

*{{cite book |title=Lecture notes on Introduction to Univalent Functions|

first=K.I|last=Noor|publisher=CIIT, Islamabad, Pakistan}}

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